Hopefully she will kill them for you. I have gone through 5 cats and not a one will kill a spider for me. But my very first kitten, she would pin down flies, chew their wings off so that the couldn't get away, and then leave their bodies laying around for me to find.
Edit: Maybe. It strongly depends on whether I step in a puddle of cat puke on the floor first thing in the morning or whether or not my black cat decides to use my bathtub as a litter box again.